Consider a flat universe with at least one finite cyclic spatial dimension: travel x meters in one direction, and you will end up back where you started.
For an object that is of small size relative to the scale of the cyclic dimension, relativistic length contraction ought to work out just fine; the cyclic nature of the space doesn't matter, and the object appears contracted in its direction of motion.
For sufficiently large objects, however, there appears to be a paradox. Consider a solid rod of length x, oriented along the cyclic dimension, so that it wraps around the universe and reconnects with itself. Topologically, it's a circle, but everywhere straight and flat. If the rod is accelerated along its length, it should appear to contract; this, however, would make it not long enough to span the cyclic space; thus, one should expect a discontinuity where two ends of the rod will break apart. There is, however, no unique location at which this discontinuity could occur.
So, what's going on? Is a flat cyclic spacetime simply not possible? Or am I missing some deeper understanding of special relativity?